go.warwick.ac.uk/SGrosskinsky/teaching/ma4h3.html
MA4H3 (2009/10)
Interacting Stochastic Processes
Problem sheet 1
1. Let W1 , W2 , . . . be a sequence of independent exponential random variables Wi ∼ Exp(λi ).
(a) Show that E(Wi ) = 1/λi and that
n
X
min{W1 , . . . , Wn } ∼ Exp
λi .
i=1
(b) The sum of iid exponentials with λi = λ is Γ-distributed, i.e.
n
X
Wi ∼ Γ(n, λ) with pdf
i=1
λn wn−1 −λw
e
.
(n − 1)!
2. The Poisson process (Nt : t ≥ 0) with rate λ > 0 is a Markov chain with X = N = {0, 1, . . .}, N0 = 0
and rates c(n, m) = λ δn+1,m .
(a) Show that Nt ∼ P oi(λt) has a Poisson distribution for all t > 0.
(b) Show that (Nt : t ≥ 0) ∼ P P (λ) if and only if it has stationary, independent increments, i.e.
Nt+s − Ns ∼ Nt − N0
and
Nt+s − Ns
independent of (Nu : u ≤ s) ,
and for each t, Nt ∼ P oi(λt).
(c) Show that for independent Poisson variables Y1 , Y2 , . . . with Yi ∼ P oi(λi ) we have E(Yi ) =
V ar(Yi ) = λi and
n
n
X
X
λi .
Yi ∼ P oi
i=1
i=1
3. The single server queue (M/M/1)
Let (ηt : t ≥ 0) be a continuous time Markov chain with state space N = {0, 1, . . .} and jump rates
c(η, η + 1) = α ,
c(η, η − 1) = β(1 − δ0,η ) .
ηt can be interpreted as the number of customers at time t, arriving at rate α > 0 and being served at rate
β > 0.
(a) Write down the master equation for this process.
(b) Show that for α > β the process is transient, i.e. ηt → ∞ a.s. as t → ∞.
Hint: Compare to an asymmetric random walk and use the strong law of large numbers.
(c) Show that for α < β the process is positive recurrent by giving its stationary distribution µ. Is the
distribution reversible?
(d) What do you think happens for α = β?
(e) Let A ∼ P P (α) be the arrival process of customers. Show that for α < β the departure process D
is also Poisson D ∼ P P (α) given that the process is stationary (this is called Burke’s theorem).
Hint: There is an elegant proof using reversibility. Alternatively, condition on the value of ηt and
show that
P at least one departure in [t, t + ∆t) = 1 − e−α∆t .
What do you think happens for α ≥ β?
4. Give a graphical construction for the linear voter model on Λ = Z with nearest neighbour interaction
p(x, y) = δx,y+1 + δx,y−1 . Look at the sample path in reversed time. How does it look like?
5. Resolve the following ’paradox’:
A single continuous-time random walker on Z does not have a stationary distribution, but an IPS of many
random walkers has!
6. A generic algorithm to simulate continuous-time IPS is called random sequential update.
Consider the TASEP with p = 1, q = 0 on ΛL = Z/LZ (periodic boundary conditions).
Claim: To simulate (or construct a sample path of) the process do the following:
• Pick a site x ∈ ΛL uniformly at random;
• update your time counter t 7→ t + ∆t by ∆t ∼ Exp(L)
(independently each time);
• if η(x) = 1 and η(x + 1) = 0 move the particle, i.e. put η(x) = 0, η(x + 1) = 1
(+ to be understood modulo L for periodic boundary conditions);
then start over again.
(a) Show that the number of timesteps k it takes for a given particle to attempt a jump is a geometric
random variable k ∼ Geo(1/L) with mean L.
Pk
(b) Show that the waiting time until a jump attempt t = i=1 ∆ti is exponential t ∼ Exp(1), where
the ∆ti are iid realizations of ∆t for each time step.
(c) For large system size L, time increments are often replaced by their mean for simplicity, i.e. ∆t =
1/L. Show that in this case the waiting time t = k ∗ ∆t is still exponential in the limit L → ∞.
(d) How does this algorithm have to be modified to simulate the ASEP with p, q > 0?
7. (Hard) part of the first exam question 2008/09:
We have seen that the condition on the jump rates
sup
y∈Λ
X
x∈Λ
sup c(x, y, η) < ∞
η∈X
implies (but is not equivalent to) Lf being a convergent sum for all cylinder functions f .
(a) Which of the following conditions
(i) sup sup sup c(x, y, η) < ∞
y∈Λ x∈Λ η∈X
(ii)
XX
y∈Λ x∈Λ
sup c(x, y, η) < ∞
η∈X
has the same implication, i.e. Lf converges for all cylinder functions f .
Justify your answer by using the above result or by giving an example such that Lf diverges.
(b) Suppose that the rates are translation invariant and of finite range R > 0, i.e. for all x, y ∈ Z
c(x, y, η) = c(0, y − x, τ−x η) ,
c(x, y, η) = 0 if |x − y| > R
and c(x, y, η z ) = c(x, y, η) for all z ∈ Z such that |x − z|, |y − z| > R.
Show that Lf converges for all cylinder functions f .